Calculus for History Majors

As we discover the huge role complex natural systems have in change of all kinds, we’re finding that evolving systems are our environment, the whole context and much of the shape of history.

It’s high time history majors learned about the best method available for reading their changes. A most curious and revealing thing about complex systems is that the first evidence of emergent change is often a display of the physical property that corresponds to the central mathematical idea of calculus, continuity.

In a mathematical function you can define a slope, and the same is true of almost any real change in complex systems. Complex systems evolve through progressions, and applying a logic like that of calculus to measures of change over time shows you where the progressions emerge from the noise and when they shift.

It reveals a great deal about the nature of a system because it provides direct evidence of it’s creative behavior as a whole.

That has never been the reason for teaching calculus, but it should be. The usual reason for teaching calculus is to give students their first (after 9th grade geometry) emersion experience in rigorous mathematical thinking. For history majors a little taste of that would give insight into the history of ideas, but it would give them little of use for understanding the world around them. Natural systems are not snapped together out of perfectly fitting parts like a mathematical proof is. They go through periods of eventful and uneventful change, well, like the history of civilizations, climates, ecologies, languages and life in general do.

The basic partly mathematical question is when can you look at a series of dots, and call it a curve? When can you say change has shape? If it has shape, it likely involves a complex system, and you can read the dynamics of the shape to identify change in the system’s internal structures. There is no single test, of course, since no series of dots gives a definitive description of anything, especially not a complex system.

It’s a kind of forensic exercise, finding shapes in the data, and clues in the shape that can validate them, often taking special note of periods of growth or decay, isolating the central continuity by reading through the noise and fluctuations, to find the consistent progressions that hold up to scrutiny..

In a sense what historians will be doing with a tool like this is original systems-physics research on the subjects in which they are immersed, since natural systems are essentially locally original worlds of internal physical relationships, unique separate universes. Some complex natural systems might be amenable to mathematical description, but surely not most.

What they’re more amenable to is story telling, which can be very usefully grounded in fact by directly reading the shapes of their evolutionary events, using the underlying ideas of calculus. It’s just calculus reshaped a little for exploring things beyond mathematics in the new world.